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A197184 Triangle of polynomial coefficients of the polynomial factors defined in A074051. 1
1, -1, 1, -1, -1, 1, 7, -2, -1, 1, -13, 12, -3, -1, 1, -17, -22, 18, -4, -1, 1, 199, -45, -35, 25, -5, -1, 1, -605, 465, -84, -53, 33, -6, -1, 1, -225, -1449, 910, -133, -77, 42, -7, -1, 1, 11703, -864, -3094, 1594, -190, -108, 52, -8, -1, 1, -59317, 33780, -1380, -6027, 2583, -252, -147, 63, -9, -1, 1, 83143, -179398, 78567, -771, -10899, 3948, -315, -195, 75, -10, -1, 1, 991671, 271073, -461978, 159115, 2882, -18546, 5764, -374, -253, 88, -11, -1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

The triangle T(n,k), 0<=k<n, shows the coefficients [x^k] of the polynomial p_n(x) which distributes sum_{i=1..m} i^n*(i+1)! = A074052(n) + A074051(n)*sum_{i=1..m} (i+1)! + p_n(m) *(m+2)!.

LINKS

Table of n, a(n) for n=1..91.

FORMULA

A074052(n) + 2*A074051(n) + 6*p_n(1) = 2. - R. J. Mathar, Oct 13 2011

(x+2)*p_n(x)-p_n(x-1) = x^n-A074051(n). - R. J. Mathar, Oct 13 2011

Conjectures on p_n(x)= sum_{k=0..n-1} T(n,k)*x^k:

T(n,n-1) = 1.

T(n,n-2) = -1.

T(n,n-3) = -(n-2).

T(n,n-4) = A055998(n-2).

T(n,n-5) = -(n-2)*(n^2-4*n+21)/6.

T(n,n-6) = (n-5)*(n-2)*(n^2-19*n-24)/24.

EXAMPLE

1;   1

-1,1;  -1+x

-1,-1,1;  -1-x+x^2

7,-2,-1,1;  7-2*x-x^2+x^3

-13,12,-3,-1,1;  -13+12*x-3*x^2-x^3+x^4

-17,-22,18,-4,-1,1;   -17-22*x+18*x^2-4*x^3-x^4+x^5

CROSSREFS

Sequence in context: A177969 A021585 A103713 * A089129 A100957 A191856

Adjacent sequences:  A197181 A197182 A197183 * A197185 A197186 A197187

KEYWORD

sign,tabl

AUTHOR

R. J. Mathar, Oct 11 2011

STATUS

approved

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Last modified November 19 16:10 EST 2017. Contains 294936 sequences.